Is the gradient at a layer independent of the activations of the previous layers?
Yes, this is the premise of back-propagation, the gradient at layer $j_{n}$ is not impacted by the gradient at layer $j_{n-1}$. This allows you to start with a gradient at the output layer and propagated it back through the network to the input layer.

It is however impacted by the gradient at $j_{n+1}$, which the back-prop algorithm also follows.